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Roberts’ example and the AR-property of convex sets in non-locally convex linear metric spaces. (English) Zbl 1003.54501
The paper begins with some still open, well-known problems in topology (the AR-problem, Schauder’s conjecture and the Banach problem of whether every infinite-dimensional compact convex set in a linear metric space is homeomorphic to the Hilbert cube). Then, in Section 2 the following characterization of ANR-spaces is given: A metric space is an ANR if and only if there is a zero sequence of open covers \(\{{\mathcal U}_n\}\) of \(X\) together with a map \(f:{\mathcal K}({\mathcal U})\to X\) such that, for any map \(g:{\mathcal U}\to X\) satisfying \(g(U)\in U\), \(U\in{\mathcal U}\), and for any sequence \(\{\sigma_k\}\subset{\mathcal K}({\mathcal U})\) with \(n(\sigma_k)\) converging to \(0\), we have \(\text{diam}\{g(\sigma^0_k)\cup f(\sigma_k)\}\to 0\). Here, \(\{{\mathcal U}_n\}\) is a zero sequence if \(\text{mesh}({\mathcal U}_n)\to 0\), \({\mathcal U}= \bigcup\{{\mathcal U}_n; n\in\mathbb{N}\}\) and \({\mathcal K}({\mathcal U})= \bigcup\{N({\mathcal U}_n\cup{\mathcal U}_{n+1})\); \(n\in\mathbb{N}\}\), where \(N({\mathcal U}_n\cup{\mathcal U}_{n+1})\) is the nerve of the cover \({\mathcal U}_n\cup{\mathcal U}_{n+1}\) equipped with the Whitehead topology. Also, \(n(\sigma)= \sup\{n\in\mathbb{N}; \sigma\in N({\mathcal U}_n\cup{\mathcal U}_{n+1})\}\) for each \(\sigma\in{\mathcal K}({\mathcal U})\). In Section 3 some results and questions concerning Roberts’s examples of compact convex sets with no extreme points are reviewed. The last section is devoted to the following theorem: Let \((X,\mu)\) be a normalized measure space and let \(\phi\) be an Orlicz function. Then \(L_\phi(\mu)= \{f: X\to\mathbb{R}\), \(\int_X\phi(|f|)d\mu< \infty\}\) with the \(F\)-norm \(\|f\|= \int_X\phi(|f|) d\mu\) contains a needle point space if and only if \(\mu\) is not purely atomic.
Reviewer’s comment: It is shown that the above-mentioned characterization of ANR’s (Theorem 2.1 in the paper under review) implies the Dugundji theorem that every convex set in a locally convex space is an AR. But it should be noted that in the authors’ proof of Theorem 2.1 some results from the monograph by S. Hu [Theory of retracts (1965; Zbl 0145.43003)] are used and the proof of these results (as given in Hu’s book) are based on the same technique developed by Dugundji for proving his famous theorem.
54E35 Metric spaces, metrizability
46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)